Geometry Midterm REVIEW (Units 1

Name: ______________________
Class: _________________
Date: _________
ID: A
Geometry Midterm REVIEW (Units 1 - 4)
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. The steps in the construction result in a line (m) through the given point (A) that is parallel to the
given line (n).
Which statement justifies why the constructed line is parallel to the given line?
a.
b.
c.
d.
When two lines are intersected by a transversal and corresponding angles are
congruent, the lines are parallel.
When two lines are each perpendicular to a third line, the lines are parallel.
When two lines are each parallel to a third line, the lines are parallel.
When two lines are intersected by a transversal and alternate interior angles are
congruent, the lines are parallel.
1
Name: ______________________
ID: A
Use the composition of a 90° counterclockwise rotation about C, followed by a reflection
across line l, (R l û r (90, C) )( ABC) = DFE, shown below.
____
____
2. Which side has an equal measure to CB?
a.
AB
c.
DE
b.
EF
d.
DF
c.
d.
m∠F
m∠E
3. Which angle has an equal measure to m∠B?
a. m∠A
b. m∠D
2
Name: ______________________
ID: A
Use the figures below.
____
4. Write a sequence of rigid motions that maps RST to DEF.
a. (R y = x û r (180°, P) )( RST) = ( DEF)
c. R y = x ( RST) = ( DEF)
b.
____
r (180°, P) ( RST) = ( DEF)
d.
(R ST û r (270°, P) )(
RST) = (
DEF)
5. Write a sequence of rigid motions that maps AB to XY.
a.
(T < −1, 0 > û r (90°, P) )(AB) = (XY)
c.
(T < 0, −1 > û r (90°, P) )(AB) = (XY)
b.
R x = 0 (AB) = (XY)
d.
(R y = x û r (90°, P) )(AB) = (XY)
____
6. Is the line through points P(7, –8) and Q(1, –1) perpendicular to the line through points R(–3, 4) and
S(4, 10)? Explain.
a. Yes; their slopes have product –1.
b. Yes; their slopes are equal.
c. No; their slopes are not equal.
d. No, their slopes are not opposite reciprocals.
____
7. Plans for a bridge are drawn on a coordinate grid. One girder of the bridge lies on the line y = 5x + 9. A
perpendicular brace passes through the point (3, –9). Write an equation of the line that contains the
brace.
1
a. y + 9 = 5(x – 3)
c. y + 9 = − (x – 3)
5
1
b. y + 3 = (x – 9)
d. x + 9 = 5(y – 3)
5
3
Name: ______________________
____
ID: A
8. Which of these transformations appear to be a rigid motion?
(I) parallelogram EFGH → parallelogram XWVU
(II) hexagon CDEFGH → hexagon YXWVUT
(III) triangle EFG → triangle VWU
a.
____
I only
b.
I and II only
c.
II and III only
d.
I, II, and III
9. Find the area of the quadrilateral formed by the vertices A(0, 0), B (4, -3), C(7, 1) and D(3, 4).
a. 12 units2
b. 28 units2
c. 16 units2
d. 25 units2
____ 10. Each unit on the map represents 5 miles. What is the actual distance from Oceanfront to Seaside?
a.
about 8 miles
b.
about 50 miles
c.
4
about 10 miles
d.
about 40 miles
Name: ______________________
ID: A
____ 11. Find the point Q along the directed line segment from point R(-3, 3) to point S(6, -3) that divides RS
into the ratio 2:1.
a. Q(-1, 3)
b. Q(3, -1)
c. Q(1.5, 1)
d. Q(0, 1)
____ 12. Which is a correct two-column proof?
Given: ∠H and ∠C are supplementary.
Prove: j Ä l
a.
Statements
Reasons
∠H and ∠C are supplementary.
Given
∠H ≅ ∠E
Alternate Exterior Angles
∠G and ∠A are supplementary.
Substitution
j Ä l
Same-Side Interior Angles Converse
Statements
Reasons
∠H and ∠C are supplementary.
Given
∠H ≅ ∠E
Vertical Angles
∠E and ∠C are supplementary.
Substitution
j Ä l
Same-Side Interior Angles Converse
Statements
Reasons
∠H and ∠C are supplementary.
Given
∠H ≅ ∠E
Vertical Angles
∠E and ∠C are supplementary.
Same-Side Interior Angles
j Ä l
Same-Side Interior Angles Converse
b.
c.
d.
none of these
5
Name: ______________________
ID: A
____ 13. If ΔMNO ≅ ΔPQR, which of the following can you NOT conclude as being true?
a.
∠M ≅ ∠P
b.
MN ≅ PR
c.
NO ≅ QR
d.
∠N ≅ ∠Q
____ 14. Find the glide reflection image of the black triangle for the composition (R x = 1 û T < 0, −7 > ). A translation,
followed by a reflection.
a.
c.
b.
d.
Short Answer
15. A dilation has center (0, 0). Find the image of B(18, 36) if the scale factor is
3
.
6
16. Is ΔTVS scalene, isosceles, or equilateral? The vertices are T(1, 1), V(4, 0), and S(2, 4).
6
Name: ______________________
ID: A
The polygons are similar, but not necessarily drawn to scale. Find the value of x.
17.
18.
19. Find the values of the variables in the parallelogram. The diagram is not to scale.
20. In a diagram of a landscape plan, the scale is 1 cm = 10 ft. In the diagram, the trees are 2.8 centimeters
apart. How far apart should the actual trees be planted?
7
Name: ______________________
ID: A
21. Michele wanted to measure the height of her school’s flagpole. She placed a mirror on the ground 48 feet
from the flagpole, then walked backwards until she was able to see the top of the pole in the mirror. Her
eyes were 5 feet above the ground and she was 12 feet from the mirror. Using similar triangles, find the
height of the flagpole to the nearest tenth of a foot.
22. Given ΔQRS ≅ ΔTUV, QS = 5v + 4, and TV = 9v − 8, find the length of QS and TV.
23. Use the information in the diagram to determine the height of the tree. The diagram is not to scale.
24. Is the line through points P(–6, 2) and Q(–1, –9) parallel to the line through points R(4, 4) and
S(–3, 5)? Explain.
25. The diagram below shows
what is m∠WXR?
WXY. Point R lies on WY. Based on the angle measures in the diagram,
8
Name: ______________________
ID: A
26. For the parallelogram, if m∠2 = 4x − 19 and m∠4 = 3x − 10, find m∠3. The diagram is not to scale.
27. The vertices of a triangle are P(–1, –5), Q(–5, –3), and R(–3, 7). Name the vertices of R y = x ( PQR).
Which theorem or postulate proves the two triangles are similar?
28.
29. Find values of x and y for which ABCD must be a parallelogram. The diagram is not to
scale.
30. Describe the resulting transformation that occurs after the composition of transformations
R x = 7 û R x = 3 (AB).
9
Name: ______________________
ID: A
31. How many lines of symmetry does the figure have?
32. m∠2 = 30. Find m∠4.
33. Brenda was sitting in row 7, seat 4 at a soccer game when she discovered her ticket was for row 9, seat 9.
Write a rule to describe the translation needed to put her in the proper seat.
The hexagon GIKMPR and ΔFJN are regular. The dashed line segments form 30° angles.
34. What is r (240°, O) (P)?
35. Find the angle of rotation about O that maps QR to LM.
36. A carnival ride is drawn on a coordinate plane so that the first car is located at the point (30, 0). What
are the coordinates of the first car after a rotation of 270° about the origin?
10
Name: ______________________
ID: A
37. Which statement justifies the conclusion that if b and c are both perpendicular to a, then they are
parallel?
Are the polygons similar? If they are, write a similarity statement and give the scale factor.
38.
What are the values of a and b?
39.
11
Name: ______________________
ID: A
40. Which triangles are congruent by ASA?
41. Campsites F and G are on opposite sides of a lake. A survey crew made the measurements shown on the
diagram. What is the distance between the two campsites? The diagram is not to scale.
42. Find m∠P. The diagram is not to scale.
43. What is the most precise name for quadrilateral ABCD with vertices A(–5, –5), B(–5, –2), C(–1, –2), and
D(–1, –5)?
44. What is the equation in point-slope form for the line parallel to y = 6x + 9 that contains P(–2, –5)?
12
Name: ______________________
ID: A
45. A transversal crosses two parallel lines.Which statement should be used to prove that the measures
of angles 1 and 5 sum to 180°?
46. Find the value of x for which l is parallel to m. The diagram is not to scale.
State whether the triangles are similar. If so, write a similarity statement and the postulate
or theorem you used.
47.
48.
13
Name: ______________________
ID: A
49. Justify the last two steps of the proof.
Given: MN ≅ PO and MO ≅ PN
Prove: ΔMNO ≅ ΔPON
Proof:
1. MN ≅ PO
2. MO ≅ PN
3. NO ≅ ON
4. ΔMNO ≅ ΔPON
1. Given
2. Given
3. ?
4.
?
50. What is a rule that describes the translation ABCD → A ʹ′B ʹ′C ʹ′D ʹ′?
51. Find the length of the midsegment. The diagram is not to scale.
14
Name: ______________________
ID: A
52. The vertices of a rectangle are R(–5, –5), S(–1, –5), T(–1, 1), and U(–5, 1). A translation maps R to the
point (–9, –1). Find the translation rule and the image of U.
53. Describe the resulting transformation that occurs after the composition of transformations R m û R l .
54. What is the value of x?
15
Name: ______________________
ID: A
55. The dashed-lined figure is a dilation image of the solid-lined figure. What is the scale factor of the
dilation?
56. Find the perimeter of ΔABC with vertices A(–6, –8), B(2, –8), and C(–6, –2).
16
Name: ______________________
57. A dilation maps
Find A' C' .
ABC onto
ID: A
A' B' C' . AB = 24 ft, AC = 28 ft, BC = 20 ft, and A' B' = 4 ft.
58. What is the value of x, given that PQ Ä BC?
59. Which diagram below shows a correct mathematical construction using only a compass and a
straightedge to bisect an angle?
60. Which construction is shown in the accompanying diagram?
61. Find the coordinates of the midpoint of the segment whose endpoints are H(6, 15) and K(8, 9).
62. The measures of the three angles of a triangle are given below. Find the value of x.
m∠A = (x + 10)°, m∠B = (x − 20)°, and m∠C = (x + 25)°
63. The vertices of a triangle are P(–6, –3), Q(–3, 7), and R(2, –2). Name the vertices of R y = 0 ( PQR).
17
Name: ______________________
ID: A
64. Plots of land between two roads are laid out according to the boundaries shown. The boundaries between
the two roads are parallel. What is the length of Plot 3 along Cheshire Road?
65. Find the values of x and y.
18
Name: ______________________
ID: A
66. The homeowner’s association at the Sunshine City housing complex informs Eric that the tree in his
front yard is too tall and needs to be cut down. Zoning laws state that trees cannot exceed 20 feet in
height. To measure the height of the tree, Eric stands near the tree. He measures the shadow of the tree
1
on the ground and then measures his own shadow. Eric’s shadow is 8 feet. The tree’s shadow is 32 feet.
2
1
Eric is 5 feet tall.
2
How tall is the tree?
67. If ON = 8x − 4, LM = 7x + 9, NM = x − 9, and OL = 2y − 7, find the values of x and y for which LMNO must
be a parallelogram. The diagram is not to scale.
68. State whether ΔABC and ΔAED are congruent. Justify your answer.
69. The zoom feature on a camera lens allows you dilate what appears on the display. When you change
from 100% to 400%, the new image on your screen is an enlargement of the previous image with a
scale factor of 4. If the new image is 28 millimeters wide, what was the width of the previous image?
19
Name: ______________________
ID: A
70. If the figure has rotational symmetry, find the angle of rotation about the center that results in an image
that matches the original figure.
20
ID: A
Geometry Midterm REVIEW (Units 1 - 4)
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
A
B
C
D
A
A
C
B
D
D
B
B
B
D
SHORT ANSWER
15.
B' (9, 18)
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
isosceles
16
x = 13
x = 27, y = 55, z = 98
28 ft
20 ft
19
75 ft
No; the lines have unequal slopes.
52°
163
P ʹ′(−5, − 1), Q ʹ′(−3, − 5), R ʹ′(7, − 3 )
AA Postulate
x = 5, y = 3
a translation of 8 units to the left
8
30
T <2,5> (Brenda)
34. G
35. 120°
1
ID: A
36. (0, −30)
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
If two lines are perpendicular to the same line, then they are parallel.
The polygons are not similar.
a = 4.5, b = 7.5
ΔVTU and ΔHGF
42.3 m
29º
rectangle
y + 5 = 6(x + 2)
Angles 1 and 8 are congruent as corresponding angles; angles 5 and 8 form a linear pair.
70
The triangles are not similar.
ΔOMN ∼ ΔOJK; SAS∼
Reflexive Property of ≅; SSS
T < 3, 6 > (ABCD)
51. 33
52. T < −4, 4 > (RSTU); (−9, 5)
53. a clockwise rotation of 170°
54. 5
55.
21
4
56. 24 units
57. 4
2 ft
3
58. 12
59.
60.
61.
62.
63.
the perpendicular bisector of AB
(7, 12)
x = 55
P ʹ′(−6, 3), Q ʹ′(−3, − 7), R ʹ′(2, 2)
64. 60 yards
65. x = 31, y = 18
66. 20.7 feet
11
67. x = 13, y =
2
68. yes, by either SSS or SAS
69. 7 mm
2
ID: A
70. 120°
3

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